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# Abstract algebraic logic : an introductory textbook / Josep Maria Font.

Material type: BookPublisher: London : College Publications, c2016Description: xxix, 521 p. ; 24 cm.ISBN: 9781848902077.Other classification: 03-01 (03G27)
Contents:
Detailed contents -- Short contents vii -- Detailed contents ix -- A letter to the reader xv -- Introduction and Reading Guide xix -- Overview of the contents xxii -- Numbers, words, and symbols xxvi -- Further reading xxvii -- i Mathematical and logical preliminaries  -- 1.1 Sets, languages, algebras  -- The algebra of formulas  -- Evaluating the language into algebras  -- Equations and order relations  -- Sequents, and other wilder creatures  -- On variables and substitutions  -- Exercises for Section 1.1  -- 1.2 Sentential logics  -- Examples: Syntactically defined logics  -- Examples: Semantically defined logics  -- What is a semantics?  -- What is an algebra-based semantics?  -- Soundness, adequacy, completeness  -- Extensions, fragments, expansions, reducts  -- Sentential-like notions of a logic on extended formulas  -- Exercises for Section 1.2  -- 1.3 Closure operators and closure systems: the basics  -- Closure systems as ordered sets  -- Bases of a closure system  -- The family of all closure operators on a set  -- The Frege operator  -- Exercises for Section 1.3  -- 1.4 Finitarity and structurality  -- Finitarity  -- Structurality  -- Exercises for Section 1.4  -- 1.5 More on closure operators and closure systems  -- Lattices of closure operators and lattices of logics  -- Irreducible sets and saturated sets  -- Finitarity and compactness  -- Exercises for Section 1.5  -- 1.6 Consequences associated with a class of algebras  -- The equational consequence, and varieties  -- The relative equational consequence  -- Quasivarieties and generalized quasivarieties 64 Relative congruences  -- The operator U  -- Exercises for SecHon 1.6  -- 2 The first steps in the algebraic study of a logic  -- 2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic  -- Exercises for Section 2.1  -- 2.2 Implicative logics  -- Exercises for Section 2.2  -- 2.3 Filters  -- Hie general case  -- The implicative case  -- Exercises for Section 2.3  -- 24 Extensions of the Lindenbaum-Tarski process  -- Implication-based extensions  -- Equivalence-based extensions  -- Conclusion  -- 2.5 Two digressions on first-order logic  -- The logic of the sentential connectives of first-order logic  -- The algebraic study of first-order logics  -- 3 The semantics of algebras  -- 3.1 Transformers, algebraic semantics, and assertional logics  -- Exercises for Section 3.1  -- 3.2 Algebraizable logics  -- Uniqueness of the algebraization: the equivalent algebraic semantics  -- Exercises for Section  -- 3.3 A syntactic characterization, and the Lindenbaum-Tarski process again  -- Exercises for Section 3.3  -- 3.4 More examples, and special kinds of algebraizable logics  -- Finitarity issues  -- Axiomatization  -- Regularly algebraizable logics  -- Exercises for Section 3.4  -- 3.5 The Isomorphism Theorems  -- The evaluated transformers and their residuals  -- The theorems, in many versions  -- Regularity  -- Exercises for Section 3.5  -- 3.6 Bridge theorems and transfer theorems  -- The classical Deduction Theorem  -- The general Deduction Theorem and its transfer  -- The Deduction Theorem in algebraizable logics and its applications  -- Weak versions of the Deduction Theorem  -- Exercises for Section 3.6  -- 3.7 Generalizations and abstractions of algebraizability  -- Step 1: Algebraization of other sentential-like logical systems  -- Step 2: The notion of deductive equivalence  -- Step 3: Equivalence of structural closure operators  -- Step 4: Getting rid of points  -- 4 The semantics of matrices  -- 4.1 Logical matrices: basic concepts  -- Logics defined by matrices  -- Matrices as models of a logic  -- Exercises for Section 4.1  -- 4.2 The Leibniz operator  -- Strict homomorphisms and the reduction process  -- Exercises for Section 4.2  -- 4.3 Reduced models and Leibniz-reduced algebras  -- Exercises for Section 4.3  -- 4.4 Applications to algebraizable logics  -- Exercises for Section 4.4  -- 4.5 Matrices as relational structures  -- Model-theoretic characterizations  -- Exercises for Section 4.5  -- 5 The semantics of generalized matrices  -- 5.1 Generalized matrices: basic concepts  -- Logics defined by generalized matrices  -- Generalized matrices as models of logics  -- Generalized matrices as models of Gentzen-style rules  -- Exercises for Section 5.1  -- 5.2 Basic full generalized models, Tarski-style conditions and transfer theorems  -- Exercises for Section  -- 5.3 The Tarski operator and the Susako operator  -- Congruences in generalized matrices  -- Strict homomorphisms  -- Quotients  -- The process of reduction 266 Exercises for Section 5.3  -- 5.4 The algebraic counterpart of a logic  -- The L-algebras and the intrinsic variety of a logic  -- Exercises for Section 5.4  -- 5.5 Full generalized models  -- The main concept  -- Three case studies  -- The Isomorphism Theorem  -- The Galois adjunction of the compatibility relation  -- Exercises for Section 5.5  -- 5.6 Generalized matrices as models of Gentzen systems  -- The notion of full adequacy  -- -- 6 Introduction to the Leibniz hierarchy -- 6.1 Overview  -- 6.2 Protoalgebraic logics  -- Basic concepts  -- The fundamental set and the syntactic characterization  -- Monotonicity, and its applications  -- A model-theoretic characterization  -- The Correspondence Theorem  -- Protoalgebraic logics and the Deduction Theorem  -- Full generalized models of protoalgebraic logics and Leibniz filters  -- Exercises for Section 6.2  -- 6.3 Definability of equivalence (protoalgebraic and equivalential logics)  -- Definability of the Leibniz congruence, with or without parameters  -- Equivalential logics: Definition and general properties  -- Equivalentiality and properties of the Leibniz operator  -- Model-theoretic characterizations  -- Relation with relative equational consequences  -- Exercises for Section 6.3  -- 6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics)  -- Implicit and explicit (equational) definitions of truth  -- Truth-equational logics  -- Assertional logics  -- Weakly algebraizable logics  -- Regularly weakly algebraizable logics  -- Exercises for Section 6.4  -- 6.5 Algebraizable logics revisited  -- Full generalized models of algebraizable logics  -- On the bridge theorem concerning the Deduction Theorem  -- Regularly algebraizable logics revisited  -- Exercises for Section 6.5  -- 7 Introduction to the Frege hierarchy  -- 7.1 Overview  -- 7.2 Selfextensional and fully selfextensional logics  -- Selfextensional logics with conjunction  -- Semilattice-based logics with an algebraizable assertional companion  -- Selfextensional logics with the Uniterm Deduction Theorem  -- Exercises for Sections 7.1 and 7.2  -- 7.3 Fregean and fully Fregean logics  -- Fregean logics and truth-equational logics  -- Fregean logics and protoalgebraic logics  -- Exercises for Section 7.3  -- Summary of properties of particular logics  -- Classical logic and its fragments  -- Intuitionistic logic, its fragments and extensions, and related logics  -- Other logics of implication  -- Modal logics  -- Many-valued logics  -- Substructural logics  -- Other logics  -- Ad hoc examples  -- Bibliography  -- Indices  -- Author index  -- Index of logics  -- Index of classes of algebras  -- General index  -- Index of acronyms and labels  -- Symbol index  --
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Últimas adquisiciones 03 F677a (Browse shelf) Checked out 2020-08-31 A-9292

Incluye referencias bibliográficas (p. -496) e índices.

Detailed contents --
Short contents vii --
Detailed contents ix --
A letter to the reader xv --
Introduction and Reading Guide xix --
Overview of the contents xxii --
Numbers, words, and symbols xxvi --
i Mathematical and logical preliminaries  --
1.1 Sets, languages, algebras  --
The algebra of formulas  --
Evaluating the language into algebras  --
Equations and order relations  --
Sequents, and other wilder creatures  --
On variables and substitutions  --
Exercises for Section 1.1  --
1.2 Sentential logics  --
Examples: Syntactically defined logics  --
Examples: Semantically defined logics  --
What is a semantics?  --
What is an algebra-based semantics?  --
Extensions, fragments, expansions, reducts  --
Sentential-like notions of a logic on extended formulas  --
Exercises for Section 1.2  --
1.3 Closure operators and closure systems: the basics  --
Closure systems as ordered sets  --
Bases of a closure system  --
The family of all closure operators on a set  --
The Frege operator  --
Exercises for Section 1.3  --
1.4 Finitarity and structurality  --
Finitarity  --
Structurality  --
Exercises for Section 1.4  --
1.5 More on closure operators and closure systems  --
Lattices of closure operators and lattices of logics  --
Irreducible sets and saturated sets  --
Finitarity and compactness  --
Exercises for Section 1.5  --
1.6 Consequences associated with a class of algebras  --
The equational consequence, and varieties  --
The relative equational consequence  --
Quasivarieties and generalized quasivarieties 64 Relative congruences  --
The operator U  --
Exercises for SecHon 1.6  --
2 The first steps in the algebraic study of a logic  --
2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic  --
Exercises for Section 2.1  --
2.2 Implicative logics  --
Exercises for Section 2.2  --
2.3 Filters  --
Hie general case  --
The implicative case  --
Exercises for Section 2.3  --
24 Extensions of the Lindenbaum-Tarski process  --
Implication-based extensions  --
Equivalence-based extensions  --
Conclusion  --
2.5 Two digressions on first-order logic  --
The logic of the sentential connectives of first-order logic  --
The algebraic study of first-order logics  --
3 The semantics of algebras  --
3.1 Transformers, algebraic semantics, and assertional logics  --
Exercises for Section 3.1  --
3.2 Algebraizable logics  --
Uniqueness of the algebraization: the equivalent algebraic semantics  --
Exercises for Section  --
3.3 A syntactic characterization, and the Lindenbaum-Tarski process again  --
Exercises for Section 3.3  --
3.4 More examples, and special kinds of algebraizable logics  --
Finitarity issues  --
Axiomatization  --
Regularly algebraizable logics  --
Exercises for Section 3.4  --
3.5 The Isomorphism Theorems  --
The evaluated transformers and their residuals  --
The theorems, in many versions  --
Regularity  --
Exercises for Section 3.5  --
3.6 Bridge theorems and transfer theorems  --
The classical Deduction Theorem  --
The general Deduction Theorem and its transfer  --
The Deduction Theorem in algebraizable logics and its applications  --
Weak versions of the Deduction Theorem  --
Exercises for Section 3.6  --
3.7 Generalizations and abstractions of algebraizability  --
Step 1: Algebraization of other sentential-like logical systems  --
Step 2: The notion of deductive equivalence  --
Step 3: Equivalence of structural closure operators  --
Step 4: Getting rid of points  --
4 The semantics of matrices  --
4.1 Logical matrices: basic concepts  --
Logics defined by matrices  --
Matrices as models of a logic  --
Exercises for Section 4.1  --
4.2 The Leibniz operator  --
Strict homomorphisms and the reduction process  --
Exercises for Section 4.2  --
4.3 Reduced models and Leibniz-reduced algebras  --
Exercises for Section 4.3  --
4.4 Applications to algebraizable logics  --
Exercises for Section 4.4  --
4.5 Matrices as relational structures  --
Model-theoretic characterizations  --
Exercises for Section 4.5  --
5 The semantics of generalized matrices  --
5.1 Generalized matrices: basic concepts  --
Logics defined by generalized matrices  --
Generalized matrices as models of logics  --
Generalized matrices as models of Gentzen-style rules  --
Exercises for Section 5.1  --
5.2 Basic full generalized models, Tarski-style conditions and transfer theorems  --
Exercises for Section  --
5.3 The Tarski operator and the Susako operator  --
Congruences in generalized matrices  --
Strict homomorphisms  --
Quotients  --
The process of reduction 266 Exercises for Section 5.3  --
5.4 The algebraic counterpart of a logic  --
The L-algebras and the intrinsic variety of a logic  --
Exercises for Section 5.4  --
5.5 Full generalized models  --
The main concept  --
Three case studies  --
The Isomorphism Theorem  --
The Galois adjunction of the compatibility relation  --
Exercises for Section 5.5  --
5.6 Generalized matrices as models of Gentzen systems  --
The notion of full adequacy  --
--
6 Introduction to the Leibniz hierarchy --
6.1 Overview  --
6.2 Protoalgebraic logics  --
Basic concepts  --
The fundamental set and the syntactic characterization  --
Monotonicity, and its applications  --
A model-theoretic characterization  --
The Correspondence Theorem  --
Protoalgebraic logics and the Deduction Theorem  --
Full generalized models of protoalgebraic logics and Leibniz filters  --
Exercises for Section 6.2  --
6.3 Definability of equivalence (protoalgebraic and equivalential logics)  --
Definability of the Leibniz congruence, with or without parameters  --
Equivalential logics: Definition and general properties  --
Equivalentiality and properties of the Leibniz operator  --
Model-theoretic characterizations  --
Relation with relative equational consequences  --
Exercises for Section 6.3  --
6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics)  --
Implicit and explicit (equational) definitions of truth  --
Truth-equational logics  --
Assertional logics  --
Weakly algebraizable logics  --
Regularly weakly algebraizable logics  --
Exercises for Section 6.4  --
6.5 Algebraizable logics revisited  --
Full generalized models of algebraizable logics  --
On the bridge theorem concerning the Deduction Theorem  --
Regularly algebraizable logics revisited  --
Exercises for Section 6.5  --
7 Introduction to the Frege hierarchy  --
7.1 Overview  --
7.2 Selfextensional and fully selfextensional logics  --
Selfextensional logics with conjunction  --
Semilattice-based logics with an algebraizable assertional companion  --
Selfextensional logics with the Uniterm Deduction Theorem  --
Exercises for Sections 7.1 and 7.2  --
7.3 Fregean and fully Fregean logics  --
Fregean logics and truth-equational logics  --
Fregean logics and protoalgebraic logics  --
Exercises for Section 7.3  --
Summary of properties of particular logics  --
Classical logic and its fragments  --
Intuitionistic logic, its fragments and extensions, and related logics  --
Other logics of implication  --
Modal logics  --
Many-valued logics  --
Substructural logics  --
Other logics  --
Bibliography  --
Indices  --
Author index  --
Index of logics  --
Index of classes of algebras  --
General index  --
Index of acronyms and labels  --
Symbol index  --

MR, MR3558731

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